When you're getting hired you're just selling someone on the idea of you. You just need to create a signal that you're worth the purchase. There is nothing magic about a resume other than it's the most typical way an employee sells themselves. Other signals of your worth could be recommendations, credentials, publications, intelligent conversation, etc.

I prefer the "meet and greet" approach to job hunting. If we have a good conversation about their needs and how I'd satisfy them, I'm inclined to feel a little insulted when a resume is asked for.